Team:SDU-Denmark/Tour32

Modelling

Revealing the bits behind the simulation

An introduction
The projects aims to optimize the conditions present in our cells to maximize the production of rubber. Two pathways aid the conversion of glucose to the substrates of the prenyltransferase: the glycolysis and the MEP pathway. The glycolysis is heavily regulated and changes to this pathway will be diminished and overridden by the cell. However, overexpressing rate-limiting steps in the MEP pathway should allow for a faster conversion of substrates; thereby pulling the pyruvate out of the glycolysis and speeding up the overall rate of conversion.

The obvious questions that follow upon such a deduction are: Which steps in the MEP pathway are rate limiting? And what will happen to overall production economy if we speed up the pathway?

The answers to these questions indicate which steps in the MEP pathway would be most effective to manipulate.

The model

The model consists of four different velocity terms, depending on the reaction. The four terms are:

• Ordinary uni-uni reaction
• Ordinary bi-bi reaction
• Reversible bi-bi reaction
• Substrate inhibitory bi-bi reaction

All of the terms are given by their respective Michaelis-Menten velocity term.
Change in the concentration is given by the the ingoing terms subtracted the outgoing terms:

dc/dt = (Ingoing terms) - (Outgoing terms)

This allows a system of equations to be build with one differential equation for each step in the pathway.

The solution to the system is found using K_cat and K_m, which are constants related directly to a given enzyme and to substrate and product concentrations, respectively. Both can be established experimentally, and we found them in the literature. From these equations and constants, the maximum speed of each enzyme relative to the substrate(s) concentration, can then be determined.

Results

Figure 1 - We're interested in the largest slope (velocity) of the curve, here we see that the increase in slope stops after a certain increase in the starting velocity. A rate limiting step! Rate limiting steps:
Articles have shown the reaction catalysed by dxs to be the rate limiting step, and we asked the question: If we were to manipulate this step, which step would then be rate limiting? Simulating reaction kinetics often leads to stiff systems, and this is no exception. Traditional methods of solving differential equations therefore did not suffice, and MATLAB’s ODE15s, which is an L-stable solver, was used.

Simulations showed that the next rate limiting step would occur after an improvement in rate of conversion of 12 percent, and that this new rate limiting step was the reaction catalysed by IspG, as seen below. This result is worth keeping in mind when deciding on the design of the plasmids. Namely, if there was time to spare, then cloning IspG into the system might have further enhanced its functionality.

Repeating the simulation with the increased velocities (i.e. post-modifying of Dxs and IspG), it is seen that IspF catalysed the third rate limiting step. IspF would thus be an enzyme of interest, if the system was to be further improved.

Economy
Figure 2 - Here we see how much dxs reacts to B₁ vitamin. As it's reversible, it stops producing B₁ vitamin faster, than the cell can use it, which is indicated by the horizontal line. But still what we throw exponentially more and more out as a function of the speed of the pathway. The second question posed concerned the economy of the system. That is, does the increase in velocity through the pathway result in increased product, or does the substrates exit the pathway at an unintended point. Based on the literature, intermediates after the reaction catalysed by IspD pass directly through the pathway, while there exists a reversible exit to vitamin B₁ from DXP.

It has been implicitly established that the increase in velocity through the pathway does not exclusively represent an increased exit to vitamin B₁. Otherwise, there would be no increase in product, at all. So, the question is (in plain English) how much, exactly, can we increase the speed of the pathway before it starts leaking - before the amount diverted to the exit surpasses the increase in amount of product. This can be seen on the plot below.

It is noticeable that the increase in speed leads to an increase in the B₁ vitamin producing term. At increases up to 100 percent, the exit term is unproblematic, while it seems at increases above 200% that the reaction would become uneconomical, as growth of this term is exponential. However, this is not of concern, as such increases in speed through the pathway are far higher than what we hope to achieve.

In conclusion, the simulations showed the rate limiting steps to be the reactions catalysed by Dxs, IspG, and IspF. Furthermore, the outflow of the pathway was shown to have little effect at increases in velocity up to 100 percent, while additional increases in velocity would be less effective.

Modelling together with the team of Edinburgh

We wanted to model the bacterial growth as a natural consequence of the influx of resources into the cell and the consumption of these resources by the synthesis of proteins, ribosomes, and DNA. The model we uses was created by Tobias Bollenbach and incorporates Cooper and Helmstetter’s classical results about chromosome replication and the cell division cycle of Escherichia coli as well as Donachie’s “initiation mass” mechanism that couples protein synthesis to DNA replication and cell division.

The following dynamical system (Fig. 3) describes the population averages of the amount of protein p , DNA c , ribosomes r, and resources a per cell in an exponentially growing bacterial culture. The variable p describes all protein in the cell except for ribosomal protein. Proteins, DNA, and ribosomes are synthesized by the cells and are diluted as a result of cell divisions. In addition, resources are consumed in the synthesis of these cell constituents:

Figure 3.

Where the s-values are synthetic rates of their corresponding subscripts. These are given by new functions. p, r, c and a, are the concentration of protein, DNA, ribosomes and ribosomes. To fully determine the rest of the variables we have a system of nonlinear equations.

So far it has only been possible to solve these equations by solving the dynamics, even though we’re only interested in the steady state. So it should be possible to equate the differential equations to zero (steady state), and “just” solve that system. But so far only solving the dynamics have showed to be the only option which worked.

We solved the dynamics by first using a preprogrammed function solver in python to solve the nonlinear equations. To solve the dynamics itself we first used the implicit euler, which is a Radau IIA method of order one, as a prove of concept, implicit methods were necessary because of stiffness. Using such easy method we were able to prove the concept, but it has the same structure as the SDIRK method, which is of order maximum three. So if necessary we can improve the error by easily change the solver method.